This is a follow-up to my question here. Let $X= (A,+,*)$ be a real closed field. Let us define a constructible hierarchy relative to $X$ as follows. Let $D_0(X)=A\cup A^2 \cup \{+,*\}$. For any ordinal $\beta$, let $D_{\beta+1}(X)=Def(D_{\beta+1}(X))$. For any limit ordinal $\gamma$, let $D_\gamma(X)=\cup_{\beta<\gamma}D_\beta$.
Then my question is, what is the least ordinal $\lambda$ such that it’s guaranteed that $P(A)\cap D_\alpha(X) = P(A)\cap D_{\alpha+1}(X)$ for some $\alpha\leq\lambda$? Note that this is related to the notion of Gaps in the Constructible Universe.
